“Physical education also improves students’ mental health.”

I want to know that what is the difference between “student mental health” and “students’ mental health”.

student mental health - used more like a ‘label’ or ‘title’ (but obviously not quite the same, however I can’t think of a better way to describe it. Perhaps ‘designation’?

students’ mental health - the mental health of the students.

There’s no significant difference in usage there.

For this sentence, if i use ‘student mental health’ instead, would it be grammatically wrong?

No, it would be fine.

will it then be:
“Physical education also improve student mental health.” -s from improve and student?

No - you still need to use ‘improves’ as it relates to the noun ‘physical education’ (3rd person singular)